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So I am aware of the Clausius inequality which states the following:

  • ∮δQ/T≤0 around a cycle.

Now consider the following question:

Consider a real heat pump which in one cycle extracts heat Qc from the ground at temperature Tc and delivers heat Qh to a building at at temperature Th. Use the Clausius inequality to show that Qc < QhTc /Th.

  • The solution to this answer is as follows:

Using the Clausius inequality:

Qc/Tc - Qh/Th < 0 --> Qc < QhTc /Th

My question is how did we know that Qc/Tc - Qh/Th < 0

as opposed to Qh/Th - Qc/Tc <0 which would obviously yield a different result.

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    $\begingroup$ hint: $\delta Q$ is the energy the thermostat provides at temperature $T$ to the system satisfying the 1st law $dU = \delta Q + \delta W$. Thus $\delta Q$ is positive or negative whether the system absorbs or emits that amount of "heat", resp. $\endgroup$ – hyportnex Apr 13 '18 at 22:00
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The Qs in the equation are talking about the heat ON the engine. Since we are extracting heat at the colder reservoir, the sign there should be positive, while releasing heat at the hot reservoir means this should have a negative sign.

(Don't forger, if you flip the sign of an inequality, you must swap greater than and leas than)

Might be worthwhile mentioning that if you reversed the set up so you are extracting work from the engine, you would get the other result, but this is fine because efficiency is defined differently for a fridge vs a carnot engine. In neither case can you exceed the efficiency of a reversible engine.

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